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4.4.5.1 Mesh Independence Study

The mesh independence of the results from the FSI simulations was investigated by considering a case with steady airflow for a Reynolds number of 2.9 at 10 deg angle of attack. The deformation magnitude at node A (located as shown in Fig. 4.33) and the coefficients of drag and lift of the FSI surface were calculated over a range of different mesh resolutions to

demonstrate grid-independent results. The calculations from the FSI simulations were mesh independent for a mesh with 240,000 nodes or higher in the CFD model as shown in Fig. 4.34. This resolution was achieved with a global element size of 2 mm, a local element size at the FSI wing edge of 0.15 mm, and a structured layered mesh consisting of 20 layers distributed along the positive and the negative out-of-plane directions of the FSI surface with a bias of 30. The different mesh resolutions tested for the FSI simulation are shown in Table A.2 from Appendix II.

Figure 4.33 Location of the node A and central path along the spanwise y-direction. The node A was defined to monitor the convergence of the FSI results. A central path along the spanwise y-direction of the artificial wing was defined for the analysis of the FSI results.

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Figure 4.34 Mesh independence study for the FSI simulation. The variation of (A) the deformation at node A, (B) the coefficient of drag, and (C) the coefficient of lift of the artificial wing with respect to a change of the number of nodes in the CFD mesh only was monitored for the mesh independence study.

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4.4.5.2 Structural Aerodynamic Response of the Artificial Insect-Sized Wing

The deformation and stress distributions of the artificial insect-sized wing under steady airflow at different Reynolds number and angles of attack were calculated from the FSI

simulation. The considered Reynolds numbers were within the ranges at which the nature counterpart, namely the crane fly, typically maneuvers and hovers. The results from the FSI simulation for the artificial insect-sized wing were computed using a mesh grid with 350,000 nodes. This grid was generated with a global element size of 2.8 mm, a local element size at the FSI wing edge of 0.18 mm, and a structured layered mesh consisting of 50 layers distributed along the positive and the negative out-of-plane directions of the FSI surface with a bias of 100.

The steady state solution converged through an iterative process that computed an approximate solution to the system of algebraic equations derived from the discretization of the governing differential equations. The solution was considered converged when the calculated residuals for each cell fell below a specified convergence limit. For this investigation, the convergence criteria were set to 1×10-10_.

The deformation and stress responses of the artificial wing to aerodynamic loading for different Reynolds number at 10 deg angle of attack are shown in Figs. 4.35 and 4.36. In general, the stress and deformation distributions were similar for all Reynolds numbers and differed only in their magnitudes. The deformation increased nonlinearly along the spanwise y-direction of the artificial wing as observed in Figs. 4.35A, 4.35C, 4.35E, 4.35G, 4.36A, 4.36C, 4.36E, and 4.36G. The deformation at the base of the wing and its near surroundings was almost negligible due to the clamped condition assigned to the FE model and the stiffness provided in this area by the thicker veins. Contrarily, the maximum deformation was estimated near the tip of the wing where the stiffness was significantly lower compared to the stiffness at the root, mainly due to

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the presence of thinner structural veins. A similar behavior was observed on the static bending tests performed by Combes and Daniel [22] and Mengesha et al. [75] on different insect species. The results from the stress analysis in Fig. 4.35B, 4.35D, 4.35F, 4.36B, 4.36D, and 4.36F

showed higher stress concentrations at the root of the wing near the leading edge. It was expected for this region to experience high localized stress because it initially received and absorbed the incoming flow momentum.

From a mechanical point of view, the veins provided the dominating contribution to the structural dynamic response of the wing due to their higher stiffness and greater load-bearing capacity inherent to its cross-sectional structure. In essence, the veins were a network of beams that held together the wing and provided the main bending and torsional resistance to inertial and aerodynamic loadings. Nonetheless, the membrane, which was modelled as a shell, had a minor yet important contribution as a reinforcement layer capable of carrying membrane and bending forces.

The aerodynamic efficiency of the artificial wing was investigated by calculating the coefficients of drag and lift for different Reynolds numbers as shown in Figs. 4.37A and 4.37B, respectively.

The coefficient of drag decreased nonlinearly with Reynolds number and increased with angle of attack. The decrease of the coefficient of drag with Reynolds number could be explained by the viscous effects accentuated at the low Reynolds number and the development of the viscous boundary layer [103]. The increment of the coefficient of drag with angle of attack was accounted by the effects of flow separation that generated the so-called pressure drag at high angles of attack.

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The coefficient of lift increased with Reynolds number and angle of attack up to approximately an angle of 15 deg. This phenomenon was expected given that with high

freestream velocity and angle of attack a greater pressure difference between the upper and lower surfaces of the artificial wing was developed. The rate started to decrease at the 15 deg angle of attack especially for high Reynolds numbers. At this angle, the adverse pressure differences started to develop due to flow separation.

The aerodynamic efficiency of the artificial wing, defined as the ratio of the coefficient of lift to the coefficient of drag, increased with both Reynolds number and angle of attack up to the critical angle of attack of approximately 15 deg as shown in Fig. 4.37C. At this angle, the generation of lift increased slightly with angle of attack; however, the generation of pressure drag became predominant due to flow separation and reversed flow effects. These effects started to be accentuated at 30 deg, 20 deg, and 15 deg for a Reynolds number of 29, 150, and 290, respectively, as observed in Figs. 4.38–4.40. The consequences of these effects were that the aerodynamic efficiency started to decrease dramatically after the critical angle of attack.

Nonetheless, the aerodynamic efficiency of the artificial wing shows that the lift force exceeded the drag force even beyond the critical angle of attack for high Reynolds number flow. This behavior supports the evidence that the effects of the passive deformation mechanisms on the aerodynamic performance of flexible wings will be significant at high Reynolds number flow and wingbeat conditions as studied by different researchers [2, 19, 20, 33, 110, 111].

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Figure 4.35 Structural deformation response of the composite material artificial insect-sized wing. Deformation and von Mises stress of the artificial insect-sized wing under steady airflow for (A and B) Re=29, (C and D) Re=150, and (E and F) Re=290 and 10 deg angle of attack; and (G) deformation magnitude along a central path in the spanwise y-direction under steady airflow for different Reynolds number.

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Figure 4.36 Structural deformation response of the single material artificial insect-sized wing. Deformation and von Mises stress of the artificial insect-sized wing under steady airflow for (A and B) Re=29, (C and D) Re=150, and (E and F) Re=290 and 10 deg angle of attack; and (G) deformation magnitude along a central path in the spanwise y-direction under steady airflow for different Reynolds number.

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Figure 4.37 Aerodynamic performance of the artificial insect-sized wing. (A) Coefficient of drag variation with respect to angle of attack and Reynolds number, (B) coefficient of lift variation with respect to angle of attack and Reynolds number, and (C) aerodynamic efficiency of the artificial insect- sized wing.

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Figure 4.38 Velocity vectors for a Reynolds number of 29. Velocity vectors of the flow over the artificial wing for (a) 0 deg, (b) 10 deg, (c) 15 deg, (d) 20 deg, (e) 25 deg, (f) 30 deg, and (g) 40 deg angles of attack for a Reynolds number of 29.

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Figure 4.39Velocity vectors for a Reynolds number of 150. Velocity vectors of the flow over the artificial wing for (a) 0 deg, (b) 10 deg, (c) 15 deg, (d) 20 deg, (e) 25 deg, (f) 30 deg, and (g) 40 deg angles of attack for a Reynolds number of 150.

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Figure 4.40 Velocity vectors for a Reynolds number of 290. Velocity vectors of the flow over the artificial wing for (a) 0 deg, (b) 10 deg, (c) 15 deg, (d) 20 deg, (e) 25 deg, (f) 30 deg, and (g) 40 deg angles of attack for a Reynolds number of 290.

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